394 Mr. 0. Heaviside on Electromagnetic Waves, and the 

 because, at r=a, these make 



V.-V,-* and f.=^; . . (166) 



i. e. potential-difference e and continuity of displacement. 

 The normal component of displacement is 



_c_dVx__c_ m(m+l) . (1Q7) 



therefore, integrating over the sphere, the total work done 

 by e is 



«j=s««S#£t£ • • • • < 168 > 



w (2m + l) 2 ' v 7 



which agrees with the estimate (156), because 



* de V de /1CQ\ 



f = -oJd=aTS * • ' * < 169 > 



finds the vorticity, f, from the radial impressed force e; or, 

 taking 



.'. — — - = vorticity, 

 so that the old f m = eja. 



25. Single Circular Vortex Line. — There are some advan- 

 tages connected with transferring the impressed force to the 

 surface of the sphere, as it makes the force of the flux and 

 the force of the field identical both outside and inside. At 

 the boundary F is continuous, E discontinuous. 



Let the impressed force be a simple circular shell of radius 

 a, and strength e. Let it be the equatorial plane, so that the 

 equator is the one line of vorticity. Substitute for this shell 

 a spherical shell of strength \e on the positive hemisphere, — \e 

 on the negative, the impressed force acting radially. Expand 

 this distribution in zonal harmonics. The result is 



15 . 1 . 3 . 5 



2.4.6.8 



Q7+...I, • . (NO) 



so that we are only concerned with the odd m's. This equa- 

 tion settling the value of e m , the vorticity is 



2 e f'QL=2/ m KK, (171) 



